#97209

order	512
nilpotency class	3
isoclinism	isoclinic class of #97164
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

499 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,7,6,3,4,8)(10,14,16,19)(15,23,24,22)(25,26,29,32,30,34,28,33)(27,35,36,31,37,39,38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,16)(14,21)(17,22)(18,24)(20,23)(25,27)(26,31)(28,38)(29,36)(30,37)(32,39)(33,35)(34,40);(1,4,6,5)(2,7,3,8)(9,11,13,18)(10,15,16,24)(12,20,21,17)(14,22,19,23)(26,32)(28,29)(31,39)(33,34)(35,40)(36,38)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=h,e^2=g,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4405);